A solenoid of radius 4 cm and 1000 turns , carries a current of 8 A. If it is equivalent to a magnet of same size and magnetisation `vec(M)` (Magnetic moment / Volume) of `20000 Am^(-1)` , then the length of the solenoid is

Options

A. `10 pi cm`
B. `20 pi cm`
C. `30 pi cm`
D. `40 pi cm`

Answer

To find the length of the solenoid, we can follow these steps:

### Step 1: Understand the relationship between magnetic moment (M), current (I), number of turns (N), and length (L) of the solenoid.

The magnetic moment (μ) of a solenoid can be expressed as:
\[ \mu = I \cdot A \cdot N \]
where:
- \(I\) = current in amperes (A)
- \(A\) = cross-sectional area of the solenoid
- \(N\) = number of turns of the solenoid

### Step 2: Calculate the cross-sectional area (A) of the solenoid.

The area \(A\) of a solenoid with radius \(r\) can be calculated using the formula:
\[ A = \pi r^2 \]
Given that the radius \(r = 4 \text{ cm} = 0.04 \text{ m}\):
\[ A = \pi (0.04)^2 = \pi (0.0016) \approx 0.0050265 \text{ m}^2 \]

### Step 3: Substitute the values into the magnetic moment formula.

Now, substituting the values into the magnetic moment formula:
\[ \mu = I \cdot A \cdot N = 8 \cdot 0.0050265 \cdot 1000 \]
Calculating this gives:
\[ \mu \approx 8 \cdot 0.0050265 \cdot 1000 \approx 40.212 \text{ A m}^2 \]

### Step 4: Relate magnetization (M) to magnetic moment and volume.

Magnetization \(M\) is defined as:
\[ M = \frac{\mu}{V} \]
where \(V\) is the volume of the solenoid. The volume \(V\) can be calculated as:
\[ V = A \cdot L \]
Thus, we can express magnetization as:
\[ M = \frac{\mu}{A \cdot L} \]

### Step 5: Rearrange to find the length (L) of the solenoid.

Rearranging the equation gives:
\[ L = \frac{\mu}{M \cdot A} \]

### Step 6: Substitute the known values into the equation.

We know:
- \(\mu \approx 40.212 \text{ A m}^2\)
- \(M = 20000 \text{ A/m}\)
- \(A \approx 0.0050265 \text{ m}^2\)

Substituting these values:
\[ L = \frac{40.212}{20000 \cdot 0.0050265} \]

Calculating the denominator:
\[ 20000 \cdot 0.0050265 \approx 100.53 \]

Now, substituting back:
\[ L \approx \frac{40.212}{100.53} \approx 0.399 \text{ m} \approx 0.4 \text{ m} \]

### Step 7: Convert to centimeters.

To convert meters to centimeters:
\[ L \approx 0.4 \text{ m} = 40 \text{ cm} \]

### Final Answer:
The length of the solenoid is **40 cm**.

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